New results in extremal problems with polynomials
نویسنده
چکیده
In this paper we give some new results in extremal problems with polynomials which are generalization of some results of F. Locher. 2000 Mathematical Subject Classification: 26D15, 65D32 In [6] F. Locher studies some extremal problems for semidefinition polynomials (nenegative or nepositive) with the dominant coefficient equal to 1. For this he uses the quadrature formulae with high algebraical degree of exactness. Thus, using Gauss Jacobis quadrature formula, he proves: Proposition 1. For any polynomial p2m(x) ≥ 0, x ∈ [−1, 1], with the dominant coefficient equal to 1, the inequality 1 ∫ −1 (1− x)(1 + x)p2m(x)dx ≥ 2 ·m!Γ(α+m+ 1) (α+ β + 2m+ 1) ·
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